Equivalence of Steinness and Validity of Oka's principle for subdomains fo Stein manifolds

Abstract

K. OKA [ll] proved in 1939 that every continuously solvable COUSIN-I1 problem on a domain of holomorphy is also holomorphically solvable. This principle characterizes analytic properties topologically and was called by J.-P. SERRE OKA'S principle. I n 1957 H. GRAUERT [5] obtained a profound generalization of this principle, which asserts, in particular, that the holomorphic and the topologic classifications of vector bundles over a STEIN space coincide. In future H. GRAUERT'S result was developed in several directions (see, for example, [12], [4], [2], [8], [6], [lo]). I n the present paper we investigate the converse problem whether from the validity of OKA'B principle follows Smmmess or not. As it was remarked in [7], [9], this question was recently posed by K. SHIGA to J. KAJIWARA. A similar question with respect to punctured polydiscs recently was posed by V. P. PALAMODOV to the author.

Open sets which satisfy the Oka-Grauert principle in a Stein space

“…Kajiwara et al [23] obtained a similar result for a domain with C 1 -smooth boundary in a separable Hilbert space. On the other hand, by Leiterer [27], a domain D in a Stein manifold X of an arbitrary dimension is Stein if and only if H 1 (D, O) = 0 and H 1 D, O GL(r, C) → H 1 D, C GL(r, C) is quasi-injective for every r ∈ N. Patyi [33] obtained an infinite dimensional analog of this theorem of Leiterer [27].…”

A Reciprocal to the Oka‐Grauert Principle

“…[5], [6], [7]. The present note intends to extend a theorem of Leiterer [7] and a recent slight improvement of it [ll], by providing bounds for the rank of the vector bundles in a reciprocal to the OKA-GRAUERT principle on any finite dimensional STEIN space.…”